atlas[grad] - gradient operator 

Calling Sequence: 

    grad(expr) 

Parameters: 

       expr - any 0-form expression. 

Description: 

• The grad - procedure allows one to calculate the gradient operator on a 0-form expression. For any 0-form we have   by definition.

 

Examples: 

>	restart: with(atlas):

 

Declare constants:  

>	Constants(Lambda);

 

(2.1)

 

Declare functions:  

>	Functions(f=f(x,y),h=h(f));

 

(2.2)

 

Declare p-forms:  

>	Forms(e[k]=1,xi=1);

 

(2.3)

 

Declare vectors:  

>	Vectors(E[j],X,Y,Z);

 

(2.4)

 

Declare coframe:
Coframe(e[1]=d(x),e[2]=d(y)); 

(2.5)

 

Declare frame:
Frame(E[k]); 

(2.6)

 

Declare metric:
Metric(g=4*(d(x)&.d(x)+d(y)&.d(y))/(1+Lambda*(x^2+y^2))^2); 

(2.7)

 

Using grad procedure: 

Just definition:
'grad(z)'=grad(z); 

(2.8)

 

For function f:
'grad(f)'=grad(f); 

(2.9)

 

For h which depends on f:
'grad(h)'=grad(h); 

(2.10)

 

For 0-form - coordinate x:
'grad(x)'=grad(x); 

(2.11)

 

Some more examples:
'grad(cos(x)*sin(y)+x^2)'=grad(cos(x)*sin(y)+x^2); 

(2.12)

 

>	'grad(x*ln(y))'=grad(x*ln(y));

 

(2.13)

 

>

 

See Also:  

atlas, atlas[d], atlas[cov], atlas[div], atlas[L], atlas[dual].